Aerodynamic Coefﬁcient Modelling of Cylindrical Space

Aerodynamic Coefficient Modelling of Cylindrical Space Debris Analogues DuringAtmospheric Entry

Nathan Lee Donaldson(1)

(1)University of Oxford, Southwell Building, Osney Mead, Oxford, UK, OX2 0ES

ABSTRACT

A machine learning model for predicting the drag coefficient, CD of arbitrary cylindrical space debris items ispresented. The model is computed using Gaussian Process Regression (GPR) and is trained using a set of 30,000aerodynamic simulations performed using RAC, a hypersonic panel method code. The model accepts the freestreamMach and Knudsen numbers (Ma and Kn , respectively), the freestream vector pitch angle, α, and the cylinder aspectration, L/� as inputs, and returns a single drag coefficient for that condition.

The formulation of the GPR model and the methods utilised by RAC are described, followed by an evaluation ofthe accuracy of GPR model predictions when compared to those calculated using RAC. A test set of 1,000 randomfreestream conditions is input into both the GPR model and RAC, and the results compared using various statisticalmetrics. Excellent correlation is demonstrated between both methods, indicating that the GPR model is well conditioned.An improvement in computational efficiency is also observed, with a decrease in execution time of roughly one orderof magnitude compared to RAC, and a noticeable decrease in RAM usage.

1 INTRODUCTION

In the study of atmospheric entry, defunct satellites are often modelled using what is known as the “object-oriented”approach (see Figure 1a). This type of analysis involves constructing a numerical analogue of a re-entering spacecraftusing geometric primitives to represent its constituent components. Shapes such as cuboids, cylinders, cones, andspheres may be used to represent the likes of electronics enclosures, reaction wheels, fuel tanks, antennae, etc.,affording this type of modelling a high degree of versatility.

During a typical analysis of this type, a full model of the re-entering spacecraft is constructed using the geometricprimitives described above, and its trajectory propagated through the Earth’s atmosphere using 6 DOF (degree offreedom) dynamics. At either a pre-defined altitude or as the result of a transient heat transfer calculation, thebreakup of the spacecraft model is simulated. From this point onwards in the simulation, the geometric primitivesrepresenting the components are modelled individually, thereby facilitating the calculation of thermal loads andtouchdown coordinates (provided of course that they do not burn up during their transit through the atmosphere).

A major drawback of this method as it is currently employed, however, lies in the calculation of the debrisanalogues’ aerodynamic properties. As is mentioned above, the full model of a spacecraft is usually simulated using6 DOF dynamics, with the relevant aerodynamic derivatives being provided by hypersonic panel methods. Oncethe breakup event is simulated, however, the individual components are typically assessed aerodynamically usingtumble-averaged correlations (see Figure 1b) in order to ensure computational efficiency. Such correlations are alarge potential source of uncertainty in debris re-entry calculations, and as such limit the potential accuracy of theseotherwise extremely efficient numerical methods. An alternate strategy relies upon a panel method calculation foreach component at every step of its trajectory, which can quickly become computationally expensive (especially whenperforming uncertainty analyses).

In order to improve the accuracy of object-oriented debris re-entry codes, a series of new aerodynamic databaseshave been generated using a type of machine learning known as GPR (Gaussian Process Regression) modelling. Thesedatabases, which are currently implemented for a generic cylinder of aspect ratios in the range 0.1 6 L/� 6 10,utilise 4 dimensional covariance functions (kernels) to correlate freestream Mach and Knudsen numbers, the angleof incidence of the cylinder, and the aforementioned aspect ratio. The model is trained on a single aerodynamiccoefficient (drag, CD), with input data being generated using a hypersonic panel method analysis code called RAC(Re-entry Aerothermal Calculator). Once trained, GPR models are extremely computationally efficient, allowing themto feasibly replace tumble-averaged correlations and component-level panel methods in future debris re-entry codes.

6199.pdfFirst Int'l. Orbital Debris Conf. (2019)

Description of the model’s construction and the underlying theory of GPR methods are presented, followed by asummary of the simulations performed using RAC. The GPR model for the drag coefficient, CD, is then tested usinga random set of input variables which are also used as input data for further RAC simulations. Next, these two setsof data are compared in order to ascertain the accuracy of the GPR model’s predictions compared to RAC. Hence, theability of such GPR models to potentially improve future debris re-entry calculations is assessed.

(a) Example of objectoriented satellite model(NASA-ORSAT) [1]

(b) Kn dependant CD curves for cylinders of various aspect ratios(ESA-DRAMA) [2]

Figure 1: Examples of current object oriented debris assessment methods

2 GAUSSIAN PROCESS REGRESSION (GPR)

Gaussian process regression (also known as GPR modelling, Kriging, or Wiener-Kolmogarov prediction) is a typeof surrogate modelling that has gained a great deal of attention in machine learning research and as an optimisationtechnique for engineering problems, owing primarily to its generality and scalability. Rasmussen and Williams [3]have exhaustively described the process of formulating and exploiting GPR models as a general machine learningtool, while their use in engineering design was pioneered by Sacks et al. [4], who used them to design and correlatecomputer simulations. More recently, GPR models have been used by Forrester, Sobester, and Keane [5] as a majorcomponent in engineering optimisation problems. In the field of geostatistics, GPR models are known as Krigingmodels; named after Danie Krige, a mining engineer who first used the procedure to estimate the density of golddeposits from a small scattering of exploratory boreholes.

Initially, a GPR model is defined solely by its mean and covariance (kernel) functions; this is termed the “priordistribution” (Duvenaud [6] observed that a GPR model’s mean is often assumed to be zero, as fluctuations in themagnitude of the mean may be compensated for by adding additional terms to the kernel). Then, as with any surrogatemodel or regression method, the coordinates of an observed variable (x) are specified, as well as the value of saidvariable (y) at that point. The GPR model is then “trained” using these data by combining the prior with a Gaussianlikelihood function for each of the observed values; this is termed the “posterior”. As such, the parameters of thecovariance function at each observed point are optimised for the given data set, generating what is known as the“covariance matrix”. Once this process is complete, a prediction (y′) may be performed by calculating the weightedaverage of the model at some new point (denoted by the vector x′). This provides the best linear unbiased estimator,as well as the value of the covariance of the GPR model at x′.

Kernel functions are typically defined by a lengthscale, `, and their variance from the mean, σf . Since σf issimply a scaling factor, it is the value of ` which determines the “relevance” of the function to the value of a predictedvariable. As it is unlikely that the response of a system with multiple input parameters will depend on them all to thesame degree, a technique known as “automatic relevance determination” (ARD) is used, whereby separate covariancefunctions are utilised for different input parameters (termed “dimensions” here). These are then added or multiplied


together across the Hilbert space of the GPR model for each observation.

KRQ,ARD(x, x′) =

D∏d=1

σ2fd

(1 +

(xd − x′d)2

2αd`2d

)−αd

(1)

The kernel function, KRQ,ARD , utilised in the present work is defined as the product of 3 separate rationalquadratic (RQ) functions across D dimensions, as shown in Eq. (1). Here, the parameter α allows additionaladjustment of the kernel’s response (note that, unless otherwise stated, α is used from here on to represent pitchangle). Plots of the kernel function defined in Eq. (1) with D = 1 and D = 2 are presented in Figures 2a and 2b,respectively.

The software utilised for all of the GPR modelling in the present work is GPy, which was developed by theSheffield Machine Learning Group [7].

4 2 0 2 4X

0.2

0.4

0.6

0.8

1.0

k(X,

[[0.

0]])

(a) D = 1

3 2 1 0 1 2 3X

3

2

1

0

1

2

3

k(X,

[[0.

0, 0

.0]])

(b) D = 2

Figure 2: KRQ,ARD kernel functions with one and two input parameters (dimensions)

3 CALCULATION OF HYPERSONIC AERODYNAMICS USING PANEL METHODS

The aerodynamic calculations on which the GPR model is trained were performed using RAC [8], a hypersonic panelmethod code. Panel methods rely upon the inclination of planar surfaces with respect to the velocity vector of theoncoming flow (see Figure 3), and are generally used for the rapid assessment of aerothermodynamics on re-enteringobjects [9], or for the preliminary design of aircraft [10], missiles [11], launch vehicles [12], and re-entry capsules[13].

These methods were originally developed in the 1960s and, owing to the computational limitations of the era, wereused in lieu of the large scale finite volume CFD (computational fluid dynamics) calculations that are widely utilisedtoday.

RAC accepts a triangulated surface mesh of a re-entering object as input, and performs a series of geometriccalculations in order to formulate the surface normals and shear vectors. The code also allows the user to set a scalevector whereby the vertices that comprise a mesh may be proportionally shifted along a given axis, thereby altering theaspect ratio of the object. An example of this operation is illustrated in Figure 4. Figure 4b shows the original surfacemesh (a unit cylinder), while Figures 4a and 4c show the results of scaling along the longitudinal axis using factorsof 0.1 and 10, respectively. Euler angles may also be input by the user in order to facilitate rotation of the freestreamvector (as opposed to rotating the mesh itself in order to change the simulated object’s attitude).


TN

PAN

EL

FREESTREAM

NORMAL TANGENTIAL

V

θ

δ

Figure 3: Panel coordinate system and primary vectors of interest for RAC calculations

Once the initial processing of the geometry has been completed, RAC begins the process of panel shieldingcalculations. These are designed to identify those panels which would have flow directly incident upon them, and thosethat would not (and are therefore likely to be in a wake region, or would be obscured by a feature of the geometry).These are referred to as “windward” and “leeward” panels, respectively. In the present work, a basic shielding checkwas found to suffice, owing to the relative simplicity of the geometry being analysed. This check involved assessingwhether the local normal inclination angle, δ, was greater than 90o on a panel-by-panel basis, thereby identifyingwhich panels were facing the freestream vector (tagged as windward), and those that were not (tagged as leeward).

(a) � = 1, L = 0.1 (b) � = 1, L = 1 (c) � = 1, L = 10

Figure 4: Examples of cylinder surface mesh scaling used in RAC analyses

In order to perform its calculations, RAC first assesses the freestream Knudsen number for each set of flowconditions input into the solver using Eq. (2). These Knudsen numbers are then grouped into three ranges: continuum(Kn < 1× 10−3), transition (1× 10−3 6 Kn 6 10), or free molecular (Kn > 10). Following these calculations, theappropriate solver for each flow condition is selected.

Kn =λ

L=

kBT√2πσ2pL

=Ma

Re

√γπ

2(2)

Finally, once pressure and shear coefficients (Cp and Cτ ) have been obtained using appropriate methods, thefreestream conditions are used to calculate the pressure acting on each panel and, in conjunction with the previously


computed normal and shear vectors, the panel force vector. These forces are then summed in each direction of theobject’s coordinate system and the aerodynamic coefficients calculated. For the present work, the reference area usedin these equations was the cross-sectional area of the cylinder, i.e. Aref = L ·�.

3.1 CONTINUUM AERODYNAMICS

In the continuum regime, RAC may utilise several different methods to calculate the panel pressure coefficients, Cp,although for the present work, only the modified Newtonian method (Eq. (3)) is used for windward panels. The shearcoefficient, Cτ is assumed to be zero, owing to the high Reynolds numbers, Re, expected in this regime during re-entry(such high values of Re would indicate a highly inviscid flow where viscous effects are negligible compared to inertialforces).

Cpcont = Cp0cont sin2(θ) (3)

Leeward faces in the present work are treated with Eq. (4), which is known as the “70% vacuum” rule [14]. Thissemi-empirical expression operates on the expectation that recirculation and expansion into the wake region will drivethe surface pressure up above the bounding assumption of a full vacuum. As such, it predicts a pressure slightly abovevacuum, hence returning a pressure coefficient between Newtonian and full vacuum bounds.

Cpcont =−1Ma∞

(4)

3.2 TRANSITION AERODYNAMICS

In the transition regime, which exists between the continuum and free molecular regimes, RAC does not directlyperform calculations, but rather interpolates from bounding cases.

When a set of flow conditions is preprocessed by RAC, its Knudsen number is calculated using Eq. (2). If thisvalue lies in the transition regime as defined above, then RAC begins the process of bridging. To perform transitionbridging analyses, RAC calculates the mean free path lengths, λ, which correspond to Knudsen numbers of 0.001 and10.0. The program then queries the NRLMSIS00 atmospheric model [15] to find the altitudes at which these valuesof λ exist, thereby generating two bounding sets of freestream conditions - one at the upper bound of the continuumregime, and one at the lower bound of the free molecular regime.

Simulations are then run on the full range of attitudes using the appropriate continuum and free molecular solvers.As such, pressure and shear coefficient distributions are generated for the boundary cases.

RAC incorporates a global bridging method which utilises the sine-squared function of Wilmoth [16] (Eq. (5)) tobridge aerodynamic coefficients between the continuum and free molecular regimes. The parameters of the functionare set as a1 = 0.375 and a2 = 0.125 so that the curve’s gradient reduces at the appropriate values of Knudsen number.The force calculation routines must be run on the boundary cases prior to this calculation being invoked.

Ctrans = Ccont +

{(Cfm − Ccont

)sin2

(π[a1 + a2 log10(Kn)

])}(5)

3.3 FREE MOLECULAR AERODYNAMICS

In the free molecular regime, RAC utilises the analytical methods of Schaaf and Chambre [17] for both Cp (Eq. (6))and Cτ (Eq. (7)). Due to the nature of the flow in this regime, both windward and leeward faces may be treatedeffectively with these equations. The hypersonic speed ratio, s, is computed using Eq. (8).

Cpfm =1

s2

{[2− σN√

2s sin(θ) +

σN2

√TwT∞

]e−(s sin θ)

2

+[(2− σN )((s sin(θ))2 + 0.5) +

σN2

√πTwT∞

s sin(θ)

][1 + erf(s sin(θ))

]} (6)


Cτfm =σT cos(θ)

s√π

{e−(s sin(θ))

2

+√πs sin(θ)

[1 + erf(s sin(θ))

]}(7)

s = Ma

√γ

2(8)

3.4 SIMULATION CONDITIONS

The GPR model was trained with 30,000 RAC simulations, the conditions for which were calculated using a RandomLatin Hypercube (RLH) sampling plan. The limits applied to this sampling plan are presented in Table 1, whereafterthe NRLMSIS00 model was used to calculate the necessary freestream quantities (such as pressure, temperature,velocity, etc.).

Parameter Symbol Lower bound Upper bound ScaleMach number Ma 5 35 Linear

Knudsen number Kn 1× 10−5 1× 104 LogarithmicPitch angle α 0.1o 89.9o LinearAspect ratio L/� 0.1 100 Logarithmic

Table 1: Parameter bounds and scales used in generating input conditions for RAC simulations utilising the cylindergeometry.

4 RESULTS

Following training, the GPR model was tested at α = 0o and α = 90o for the L/� = 1 cylinder. A range offreestream conditions covering the entire training range of Mach and Knudsen numbers were utilised. The results ofthese calculations are presented in Figure 5, and show the expected sigmoidal trend ofCD with respect to Kn . A heavydependence on Ma is evident in the transition and free molecular regimes, with the highest values of CD occurring asMa → 5. As expected, the trends approach a true sigmoid as Ma →∞.

Ma

5 10 15 20 25 30 35

Kn1e-04

1e-021e+00

1e+02

CD

1.6

1.8

2.0

2.2

(a) α = 0o, L/� = 1

Ma

5 10 15 20 25 30 35

Kn1e-04

1e-021e+00

1e+02

CD

1.41.61.82.02.22.4

(b) α = 90o, L/� = 1

Figure 5: Mach-Knudsen surfaces of drag coefficient, CD, as predicted by the GPR model

In order to quantify the suitability of the GPR model for reproducing aerodynamic data, a random set of 1,000freestream conditions (Ma , Kn , α, and L/�) were input to both RAC and the GPR model (once again using theNRLMSISE00 model to compute additional required quantities). The two sets of results were then compared anderrors between them calculated using Eqs (9) through (13).


δ = 100

∣∣∣∣y − y′y

∣∣∣∣ (9)

δR = 100

∣∣∣∣ y − y′

max(y)−min(y)

∣∣∣∣ (10)

δRMS = 100

√∑Nn=1 |yn′ − yn|

2

N(11)

δNRMS(µ) =δRMS

y(12)

δNRMS(R) =δRMS

|max(y)−min(y)|(13)

The results of these error calculations are presented in Figure 6, and show maximum errors of the order of 1%.The mean percentage error, δ, and mean-normalised RMS (root mean square) error were both found to be less than1%. Normalisation of the errors to the range of the CD data was also performed (using Eqs. (10) and (13)) in order toprovide a more contextualised view of inaccuracies between the GPR model and the RAC results. These calculationsshow that, when considering the full range of returned drag coefficient results, the errors are roughly one order ofmagnitude smaller than those presented previously, with both the range normalised mean percentage error, δR, and therange normalised RMS error, δNRMS(R), being less than 0.1%.

10 4 10 3 10 2 10 1 100 101 102 103

CD relative error, CD (%)

0

200

400

600

800

No. o

f ins

tanc

es

= 0.367%, R = 0.041%, NRMS( ) = 0.514%, NRMS(R) = 0.064%, R2 = 1.0000

R

NRMS( )

NRMS(R)

R

Figure 6: Errors in CD predictions between RAC and the GPR model

Finally, the probability density of the two methods’ predictions were computed using the SciPy library [18]. Theresults of these calculations are presented in Figure 7, and show excellent correlation between the GPR and RACresults, thereby indicating that the probability of obtaining a given drag coefficient is virtually identical for bothmethods.

0 2 4 6 8CD

0.0

0.1

0.2

0.3

0.4

0.5

Prob

abilit

y de

nsity

, f(C

D) RAC

GPy

Figure 7: Probability density of CD predictions generated by RAC and the GPR model


5 CONCLUSIONS

In this paper, a machine learning model of the drag coefficients, CD,for cylindrical space debris has been described.The model, which utilised a correlation method called Gaussian Process Regression (GPR) was trained using 30,000aerodynamic calculations which were themselves performed using the RAC hypersonic panel method code. The GPRmodel was correlated with fresstream Mach and Knudsen numbers (Ma and Kn , respectively), the freestream vectorpitch angle, α, and the cylinder’s aspect ratio, L/�.

When queried with a randomised set of input conditions, excellent agreement was found between the modeland RAC, with peak errors of ∼ 1%, and RMS errors of 0.514% (when normalised to the mean CD result) and0.064% (when normalised to the full range of CD results). The probability density of each set of CD results was thencomputed, and also showed excellent correlation, thereby supporting the conclusion that the GPR model has capturedthe underlying trends and magnitudes of the training data set well.

Both RAC the GPR model were timed during execution, with the GPR model demonstrating a significant decreasein RAM usage and a decrease in execution time of approximately one order of magnitude when compared to RAC. Thisis significant since panel methods are known to be some of the most computationally efficient calculations availablefor aerodynamics. The resource usage and execution time of the GPR model is independent of mesh size, and themodel itself may be trained using hybrid data sets which include both inexpensive data (such as those generated byRAC) and expensive data (such as those derived from experiments or high fidelity simulations).

Future work will include the generation of additional models for other aerodynamic coefficients such as lift, CL,and pitching moment, Cmy . Further GPR models will also be generated for different geometric primitives such ascuboids, and will be integrated into an appropriate trajectory solver. This will then be used to test the effectiveness ofthe GPR models compared to existing tumble-averaged aerodynamic correlations.

Overall, it is clear that the GPR model has effectively reproduced the drag coefficient variation for a cylinder acrossa wide range of attitudes, Mach numbers, Knudsen numbers and aspect ratios. It is expected that models like this willbe incorporated into future debris demise codes in order to both expedite calculations, and increase their accuracy.Such improvements are expected to lead to reductions in cases where debris survive the re-entry process and endangerpersonnel and property on the ground.

6 REFERENCES

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Aerodynamic Coefﬁcient Modelling of Cylindrical Space